MiS Preprint Repository

This paper is the second of a series of articles on quantitatives estimates in stochastic homogenization of discrete elliptic equations. We consider a discrete elliptic equation on the $d$-dimensional lattice ${\mathbb{Z}}^d$ with random coefficients $A$ of the simplest type: They are identically distributed and independent from edge to edge.

On scales large w. r. t. the lattice spacing (i. e. unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric "homogenized" matrix $A_{hom}=a_{hom}$ Id is characterized by $\xi \cdot A_{\mathrm{hom}}\xi =⟨(\xi +\mathrm{\nabla }\varphi )\cdot A(\xi +\mathrm{\nabla }\varphi )⟩$ for any direction $\xi \in {\mathbb{R}}^d$, where the random field $\varphi$ (the "corrector") is the unique solution of $-{\mathrm{\nabla }}^{\ast }\cdot A(\xi +\mathrm{\nabla }\varphi )=0$ in ${\mathbb{Z}}^d$ such that $\varphi (0)=0$, $\mathrm{\nabla }\varphi$ is stationary and $⟨\mathrm{\nabla }\varphi ⟩=0$, $⟨\cdot ⟩$ denoting the ensemble average (or expectation).

In order to approximate the homogenized coefficients $A_{hom}$, the corrector problem is usually solved in a box $Q_L=[-L,L{)}^d$ of size $2L$ with periodic boundary conditions, and the space averaged energy on $Q_L$ defines an approximation $A_L$ of $A_{hom}$. Although the statistics is modified (independence is replaced by periodic correlations) and the ensemble average is replaced by a space average, the approximation $A_L$ converges almost surely to $A_{hom}$ as $L\uparrow \mathrm{\infty }$. In this paper, we give estimates on both errors. To be more precise, we do not consider periodic boundary conditions on a box of size $2L$, but replace the elliptic operator by $T^{-1}-{\mathrm{\nabla }}^{\ast }\cdot A\mathrm{\nabla }$ with (typically) $T\sim L2$, as standard in the homogenization literature. We then replace the ensemble average by a space average on $Q_L$, and estimate the overall error on the homogenized coefficients in terms of $L$ and $T$